@@ -150,7 +150,7 @@ We write $\Ab$ for category of abelian groups and for an abelian group $G$, we w
The functor $\lambda$ is a left adjoint.
\end{lemma}
\begin{proof}
The category of chain complexes is equivalent to the category $\omega\Cat(\Ab)$ of $\omega$-categories internal to abelian groups (see \cite[Theorem 3.3]{bourn1990another}) and with this identification, the functor $\lambda : \omega\Cat\to\omega\Cat(\Ab)$ is nothing but the left adjoint of the canonical forgetful functor $\omega\Cat(\Ab)\to\omega\Cat$.
The category $\Ch$ is equivalent to the category $\omega\Cat(\Ab)$ of $\oo$\nbd{}categories internal to abelian groups (see \cite[Theorem 3.3]{bourn1990another}) and with this identification, the functor $\lambda : \omega\Cat\to\omega\Cat(\Ab)$ is nothing but the left adjoint of the canonical forgetful functor $\omega\Cat(\Ab)\to\omega\Cat$.
\end{proof}
As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain complex $\lambda(C)$ admits a nice expression.